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Distributed Object Characterization

with Local Sensing by a Multi-Robot System

Golnaz Habibi, Sandor P. Fekete, Zachary Kingston, James McLurkin

Abstract This paper presents two distributed algorithms for enabling a swarm of

robots with local sensing and local coordinates to estimate the dimensions and ori-

entation of an unknown complex polygonal object, i.e., its minimum and maximum

width and its main axis. Our first approach is based on a robust heuristic of dis-

tributed Principal Component Analysis (DPCA), while the second is based on turn-

ing the idea of Rotating Calipers into a distributed algorithm (DRC). We simulate

DRC and DPCA methods and test DPCA on real robots. The result show our al-

gorithms successfully estimate the dimension and orientation of convex or concave

objects with a reasonable error in the presence of noisy data.

1 Introduction

Computing the geometric features of an object has many applications in robotics

and autonomous manufacturing. Collective transport, imaging, fitting the bounding

box, assembly and manipulation are some examples that involve object characteri-

zation. In collective transport, agents require the dimensions of the object and the

orientation to transport the object safely through narrow corridors or environments

with obstacles. Another application is to use object shape and orientation to manip-

ulate an object by a robot or by an industrial arm. If we can measure the object’s

main axes vectors (which implies the object orientation) as well as the object’s main

dimensions, fitting the bounding rectangle or bounding box is easy; the latter has its

own applications in manufacturing and pattern recognition.

Golnaz Habibi, Zachary kingston, James McLurkin

Rice University, Houston TX e-mail: [email protected],[email protected],

[email protected]

Sandor P. Fekete

TU Braunschweig, Germany e-mail: [email protected]

1

Appeared in Distributed Autonomous Robotic Systems 2016

2 Golnaz Habibi, Sandor P. Fekete, Zachary Kingston, James McLurkin

Width

Diameter

θl

C

(a) (b)

Fig. 1: (a) A polygonal object. The three key dimensions we need to measure are ob-

ject minimum width (width), object maximum width (diameter), and orientation(θ ).

The centroid is marked with a blue circle. Major axis is vector l. (b) Seven r-one

manipulator robots with grippers (black circles) [10] are moving the orange object.

Our goal is to transport the object along the path marked by the guide robots (blue

circles). In our previous work [5], we described algorithms that enable each guide

robot to compute a collision-free pose for the object at the robots location. This path

planning algorithm requires the object dimension to generate a safe path through

narrow corridors. Previously, we presented distributed controllers that use the guide

robots navigation information to control a group of manipulator robots [7]. In order

to navigate the object and avoid obstacles, manipulator robots measure the object

orientation and use it as a feedback to rotate the object during the transport. This

paper presents distributed algorithms to estimate object dimensions and orientation.

A common approach is to use Principal Component Analysis for finding the ma-

jor axes of the object by finding the eigenvector of the points on the boundary [3] or

inside the object [4, 8]. However, most of the previous work requires global sensing

and central processing [3] . See Fekete et al. [4, 8] for the use of Distributed Principal

Component Analysis in the context of a sensor grid of smart pixels with only local

information and communication, based on very limited computation; however, [4, 8]

makes use of the discretized grid geometry of the underlying sensor field, while this

paper considers robot positions at a limited number of continuous locations.

We aim to address situations in which global sensing, communication, and ge-

ometry are not readily available or too costly to implement. Multi-robot systems

offer the potential to estimate a global state by using distributed algorithms. In sit-

uations without GPS or global positioning, inferring global information from local

information is essential. We use multi-hop communications [1] to exchange local

information and local geometry in order to cooperate with other robots, avoiding

the need for a shared global coordinate frame [9].

Using distributed algorithms, we can break a complex task into simple subprob-

lems. In the same way, multi-robot systems are usually simple at the individual level,

but they collaboratively accomplish a task that cannot be achieved by a single robot.

We consider a scenario in which there are only sensors or robots around the

boundary of an object. We consider three key parameters of the object including 1)

centroid or center of geometry, 2) object minimum and maximum dimension, and 3)

object orientation (see Fig. 1a). These agents use local information and may not be

Distributed Object Characterization with Local Sensing by a Multi-Robot System 3

Fig. 2: A potential application of

object characterization in collective

transport: By computing the centroid

and orientation of the object, a swarm

of manipulator robots can maneuver

the object to valid free configurations

and safely transport the object.

fully connected. These include line-of-sight communication, so robots on different

sides of the object cannot communicate directly. Calculating these features helps

in estimating the shape of a closed region by using sensors on the corners of the

region with limited sensing. These features are also useful for manipulating a large

polygonal object by robots without prior knowledge of the object.

Our objective is to estimate object geometric features (centroid, object dimen-

sion and orientation) in order to use them in collective transport problem, assuming

a subgroup of robots attach themselves to the object [2, 12]. These robots, called

manipulator robots, are equipped with grippers [7]. These are responsible for esti-

mating the object dimensions, orientation and centroid. This information is provided

to the rest of the robots that cover the environment [5]; we call these guide robots.

Given the dimension of the object, guide robots generate a safe path from the object

to the goal. The last step is to transport the object through the path. By comput-

ing the centroid and orientation of the object, the manipulator robots can maneuver

and safely transport the object (see Fig. 2). Fig. 1b shows a result of a collective

transport. The path is generated based on the dimension of the object. During the

transport, manipulator robots rotate the object to avoid the collision. The details of

collective transport are out of the scope of this paper and are discussed in future

work. We developed two distributed algorithms for estimating the centroid of the

object in previous work [7, 6]. Here we present two algorithms for object character-

ization, which compute three key geometric properties of our polygonal object: its

minimum width, diameter, and orientation. The minimum width determines the nar-

rowest corridor the object can negotiate, assuming it is in the proper orientation. We

define the orientation as the direction that is perpendicular to the object’s minimum

width. The object diameter determines the minimum distance from an obstacle at

which it is safe to rotate the object.

Our first approach is Distributed Principal Component Analysis (DPCA). This

is an approximation, but it is easy to implement on real hardware and produces

good results for symmetric objects. Our second approach is a distributed version

(called DRC) of a Rotating Calipers algorithm [13] that computes exact geometry

if the manipulator robots are positioned at object vertices. We test our algorithms in

simulation and on hardware. Our results are promising, the algorithm successfully

estimates the dimension of most convex or concave objects.

The rest of the paper is organized as follows. The model and assumptions are ex-

plained in Section 2. We briefly describe our previous algorithm of average consen-


sus in Section 3. Our object characterization algorithms are described in Section 4.

Our results are discussed in Section 5, and we conclude the paper in Section 6.

2 Model and Assumptions

Object characterization can be divided into two steps. 1) Detect the boundary of the

object. 2) Estimate the object parameter based on the boundary points. We assume

the first is done by an existing algorithm [2, 12], and robots have already attached to

the object, ideally at the vertices of the object, which we assume to be simply con-

nected, so that there are no interior boundaries. The robots have no prior knowledge

of the shape or size of the object. A communication network is built by the robots

using inter-robot communications between nearby robots within a fixed distance d,

where d is much smaller than the size of the environment. We can model the robot’s

communications network, G = (V,E), as an undirected unit disk graph, obstructed

by the geometry of the object. Each robot constitutes a node u ∈ V , where V is the

set of all robots and E is the set of all robot-to-robot communication links. The set

Vm is the set of all manipulator robots. The neighbors of each node u are the set

of robots with unobstructed line of sight and communication range d of robot u,

denoted N(u) = v ∈ V | u,v ∈ E. We assume that G is connected and that it

contains a cycle that surrounds the object, so that a message sent by a robot to one

of its neighbors can be passed all the way around until it reaches the other neighbor.

Our robots are homogeneous and are modeled as disks, moving with the help of a

non-holonomic differential drive. Each robot u has a unique id, u.id, and is situated

at the origin of its own local coordinate frame with the x-axis aligned with its current

heading. Robots can measure the relative pose of their neighbors (See Fig. 3). Note

that the message-passing protocol used for Distributed Rotating Calipers works even

in an asynchronous manner; for easier description, we still describe the algorithm

execution as proceeding in a series of discrete rounds. While the robots actual oper-

ation is asynchronous, implementing a synchronizer simplifies analysis greatly and

is easy to implement [9].

Fig. 3: Local network geometry of robot v measured from

robot u. Buv is the bearing, the relative angle of robot’s head-

ing u from robot v. Ouv is the orientation, the relative heading

of robot v from robot u, and Ruv is the distance between two

robots.

Robot v

Robot u

Buv

Ouv

Ruv


3 Pipelined Consensus

In some parts of this paper, we need to estimate the average of values, including

the centroid. By definition, the centroid of a polygon with m vertices is the average

position of its vertices:

(xc,yc) =1

m

m

∑i=1

(xi,yi) (1)

Thanks to consensus-based algorithms, one can find the global estimate of vari-

ables such as max/min/average by continuously finding a local estimate of that

value [11]. In our previous work, we presented a pipelined consensus algorithm,

an extension of pairwise gossip-based consensus for multi-agent systems [6] to es-

timate global values including the object’s centroid as the average of the object’s

vertices. As a result, each robot estimates the average value (i.e., its centroid) in its

local coordinate; see [6] for more detail. In this approach, each agent starts a new

consensus in each round of gossiping, and stores the intermediate results for the

previous k consensus in a pipeline message with size k. After k rounds of gossiping,

the results of the first consensus are ready. In this paper, pipelined consensus is used

to estimate the object’s centroid as well as estimating the other parameters that are

essential for computing the object orientation. We will explain these key parameters

in the next section.

4 Object Characterization

In this section we present two distributed algorithms for extracting key geometric

features of the object including: object dimension (width and diameter) and object

orientation (see Fig. 1a).

4.1 Distributed Principal Component Analysis (DPCA)

Principal Components Analysis allows us to compute the orientation of the main

axis of the object using the vertices around the boundary[3]. We assume that the

robots are in heading consensus [11], i.e., they have agreed to a common heading.

This condition does not necessarily mean that the robots have the same orientation.

Instead, the robots can reach a common virtual heading alignment. In the beginning,

each robot estimates the position of the objects centroid at its reference frame by

using the pipelined consensus algorithm [6]. In the next part, we show that this is

sufficient information to estimate the object’s dimension and orientation.


4.1.1 Computing the Object Orientation

Given the position of centroid and robots, Chaudhuri et al. [3] presented an algo-

rithm to compute the object orientation θ , as follows.

tan2θ = 2∑

mi=1 (xi − xc)(yi − yc)

∑mi=1 [(xi − xc)2 − (yi − yc)2]

(2)

In this equation, (xi,yi), i = 1, ..,m is the global position of vertices on the bound-

ary of the object, and (xc,yc) is the centroid position in a global reference.

Lemma 1. Given the object’s centroid and a common heading for all robots (either

virtual or real), the object orientation θ is calculated by Equation (2) in the local

coordinate frame of each robot. The communication complexity is O(1) per robot,

i.e., each robot a constant number of messages of constant size.

Proof. We have to show that the components of Equation (2), i.e.,(xi−xc) and (yi−yc), are invariant with respect to local frames if the local frames reach the commonheading. In other words, a vector in a coordinate frame does not change when itis transformed to another coordinate frame if the axes of two frames are parallel.To show this, we use homogeneous coordinates. Assume there are two differentcoordinate frames i and j with the same orientation. If the origin of i with respectto j is (xt ,yt ,1) and these frames are parallel, the vector AB in the i-coordinate,

i.e. iAB = (xA −xB,yA −yB,1), is transformed to j-coordinates as follows.ji T is the

transformation matrix in homogeneous coordinates.

ji T =

1 0 xi j

0 1 yi j

0 0 1

,( jAB) = ( j

i T )(iAB) =

1 0 xi j

0 1 yi j

0 0 1

(

xA

yA

1

−

xB

yB

1

) (3)

( jAB) =

xA + xi j

yA + yi j

1

−

xB + xi j

yB + yi j

1

=

xA − xB

yB − yB

1

= (iAB) (4)

This gives us exactly the same vector in coordinate frame i. This holds for arbi-

trary points in space, as well as for centroid positions. Therefore, we can show the

following.

(xi − xci,yi − yci

,1) = (x′ci,y′ci,1), i = 1, ...,m. (5)

Here, (xi,yi) is the position of the robot i in the global reference, (xci,yci

) is the

estimated position of the centroid by robot i in global coordinate. (x′ci,y′ci

) is the

estimated position of the centroid in local coordinate i. This proves that vectors are

invariant under translation transformation of endpoints of the vector. The second

step is to show that Equation (2) can be computed in a distributed fashion. By ap-

plying the invariance feature of centroid vector, Equation (2) can be rewritten in

local coordinates, as follows.


tan(2θ) = 2

1m ∑

mi=1 (x

′ci)(y′ci

)1m ∑

mi=1 [(x

′ci)2 − (y′ci

)2](6)

We have inserted 1m

in both enumerator and denominator to simplify this equation.

By using the definition of the average S = 1m ∑

mi=1 si, we get

[(x′c)(y′c)] =

1

m

m

∑i=1

[x′ciy′ci

] and(

x′c2 − y′c

2)

=1

m

m

∑i=1

[x′ci

2− y′ci

2]. (7)

Equation (2) is simplified to

tan(2θ) = 2x′cy′c/x′c2 − y′c

2. (8)

Equation (8) requires consensus algorithms to estimate the averages x′cy′c and

x′c2 − y′c

2.We use our pipelined consensus (see Section 3) to compute these averages,

as well as the centroid vector on each robot (x,y) and η as the heading consensus.

This is the total of five averages, requiring a message of a constant size 5W for each

robot per round, where W is constant. Therefore, the message complexity is O(1)per robot.

4.1.2 Approximation of Diameter and Minimum Width by DPCA

Once we have estimated the orientation of the object, we calculate the minimum

width and the diameter with two leader election algorithms [9]. Computing the

diameter is straightforward: the robots run a leader election algorithm to find the

largest centroid distance. Using the triangle inequality, we see that the diameter is

bounded by twice this distance. Computing the minimum width requires a bit more

work. Fig. 4a shows how each robot u can compute its distance to the principal axis,

du = Ru sin(αu). We define vector l as a vector passing through the centroid (C) and

its orientation is θ . We also define dmax = maxmj=1 du. If the polygon is symmetric

across its principal axis and the vertices are balanced around the boundary, the ob-

ject’s centroid lies on the principal axis (l). In that case, the minimum width of the

polygon is exactly 2dmax for all robots u (Fig. 4b). Otherwise, the minimum width

of the polygon is no greater than 2dmax, which gives us an upper bound estimate of

min-width ≤ 2max(du) (Fig. 4c). DPCA has a good performance for most objects

and the straightforward implementation on physical systems.

4.2 Distributed Rotating Calipers

Tightening the bounds of our estimates requires more algorithmic machinery. We

must determine the best of a quadratic number of pairs of object vertices, or pairs

of vertices and object edges. In a centralized setting, reducing the computation time


u

C

v

w

x

t

s

z

y

Ru

du

θ

αuh

l

(a)

DPCA = Optimal

(b)

DPCA

Optimal

C le

e

e

(c)

Fig. 4: (a) Geometric computation of distance from a vertex (robot) to the main axis.

(b) DPCA illustration on minimum-width estimation. The DPCA error estimate is

zero and (c) the DPCA estimation error is bounded by 2e (top), where e is the offset

of object centroid (letter C) from the principal axis.

can be achieved with the method of Rotating Calipers. The idea was first conceived

by Shamos [13], the name coined by Toussaint [14]; the key is to keep track of a pair

of opposite tangents enclosing the object; updating the contact points during a full

rotation gives rise to a (centralized) Θ(n) algorithm for computing minimum and

maximum width, i.e., the diameter. In our distributed setting, a straightforward im-

plementation ends up being quadratic, as a single update of opposite contact points

requires long-distance communication, which may take Ω(n) communication steps.

In the following, we develop a distributed variant with overall time O(n). The key

pass messages

P

e4

e3

e5

e0

r0

ℓ′(e1)

r7

eh−1

rh−1

e6

r6 = ro(1)

e6

e1 r1

e2

r2

r3

r4

r5

ℓ(e1)

d1,6

(a)

P ri

r j

rk

γi

(b)

Fig. 5: (a) The basic setup for the algorithm DRC. The object contour is blue,

convex hull edges are shown in red. Messages are passed along the chain of robots

in order to compute and compare geometric information. (b) The aperture angle γi at

a robot ri. ri is on the (strict) convex hull if and only if γi is at most (strictly smaller

than) 180 degrees.


idea is to use pipelined communication along the perimeter of the object, with ge-

ometric updates performed on the fly, such that only the minimum and maximum

width for each object vertex and each object edge are tracked. See Fig. 5a for the

basic idea. This method yields exact results when we have accurate coordinate mea-

surements, but requires a more sophisticated overall protocol. This model also uses

the assumption that the robots can perceive any part of the object or any other robot

that can be reached by an unobstructed line of sight.

4.2.1 Convex Hull

First, each robot determines whether it lies on the convex hull by checking the angle

under which it sees the object P, based on the following lemma; see Figure 5b.

Lemma 2. A robot ri on the perimeter of P is on the (strict) convex hull, if and only

if it sees P within an aperture angle γi at most (strictly smaller than) 180 degrees.

This yields the set of h corner robots that lie on the boundary of the convex hull.

In the following, we focus on communication between corner robots; implicitly,

this may use non-hull robots as relays. We assume that, adjacent hull robots are

connected, and non-hull robots lie between precisely between two hull robots, with

direct access to other hull vertices blocked by the geometry of the object.

4.2.2 Computing Minimum Width

The minimum width of P is the the width of a narrowest corridor that can be passed

by the object. This can be evaluated as follows.

Lemma 3. Let P be a convex polygon with h vertices. Then for polygon vertices

r0, . . . ,rh−1 and polygon edges e0 = (rh−1,r0), . . . ,eh−1 = (rh−2,rh−1), the minimum

width of P is minh−1i=0 maxh−1

j=0 di, j, where di, j := d(ℓ(ei),r j) is the Euclidean distance

between the line ℓ(ei) through edge ei and r j.

The following observation is the basis for the idea of rotating calipers, i.e., paral-

lel tangents at opposite sides of the polygon: a pair of opposite sides that attains min-

imum distance induces a pair of parallel tangents, i.e., a minimum-width corridor.

For any edge ei, we denote by o(i) the corresponding “opposite” index, such that ver-

tex ro(i) is the first one after ri (in counterclockwise order) that attains maxh−1j=0 di, j.

Lemma 4. Let P be a convex polygon with h vertices. For i∗ and j∗ with

minh−1i=0 maxh−1

j=0 di, j = d(ℓ(ei∗),r j∗), there is a tangent, ℓ′((e)i∗), to P through r j∗ =

ro(i∗) that is parallel to ℓ(ei∗), such that P lies between ℓ((e)i∗) and ℓ′((e)i∗).

Based on this lemma, we describe a distributed algorithm. In the following, the

vertex description Di for a corner robot ri consists of its own coordinates (xi,yi),along with the coordinates of both of its neighbors, (xi−1,yi−1) and (xi+1,yi+1). The

angle αi of (ri,ri+1) with the x-axis and the angle βi of (ri,ri−1) with the x-axis can

be deduced from this information; they describe the visibility cone in which robot

ri sees P. (In a practical setting, it is easiest to simply measure these angles, rather


than computing them by means of trigonometry.) Originally, Di is unappended, if

it contains only the vertices; it is appended and denoted by D∗i , if it also contains

an “enclosure bit”, i.e., the information by robot ro(i)) opposite to edge ei that the

parallel tangents ℓ(ei) to P through (ri−1,ri) and ℓ′(ei) through ro(i) enclose P, along

with distance di,o(i) between those tangents. Overall, the smallest of these distances

is computed as follows.

Distributed Rotating Calipers (DRC)

(1) Any robot checks whether it is a corner robot by considering its visibility cone.

(2) Elect a leader corner robot, r0, as the one with the smallest ID.

(3) By passing a message from r0 around the hull, establish the (counterclockwise)

cyclic order of corner robots along the hull; let this be r0, . . . ,rh−1,r0, such that

each corner robot knows its predecessor and successor. This also determines the

hull edges, e0, . . . ,eh−1.

(4) Pass around vertex descriptions, as follows.

(4.1) All robots start in “unappended” mode.

(4.2) Robot r0 begins with sending its own (unappended) D0 to robot r1.

(4.3) While in “unappended” mode, a robot r j:

• based on angle information, checks for any incoming unappended Di (orig-

inating from some robot ri 6= r j) whether the line parallel to ℓ(ei) through

r j separates r j−1 from r j+1, i.e., whether the angle of (ri,ri−1) with the

x-axis lies between α j and β j;

• if not, then r j is a robot furthest from the line ℓ(ei), i.e., j = o(i), and Di is

appended with di, j, turning Di into D∗i ;

• passes on Di or D∗i (whether appended or not) to its successor;

• upon receiving D j−1 from its predecessor r j−1, passes it on to its successor

r j+1, followed by its own (newly minted) D j in the next round;

• upon receiving its own D∗j , switches into “appended” mode.

(4.4) While in “appended” mode, a robot r j:

• keeps track of the smallest encountered di,o(i) for a D∗i ;

• when receiving D∗0 for the second time, passes on D∗

0, then STOPs;

• passes on any received D∗i .

We claim the following; note that bookkeeping applies to the convex hull ver-

tices, with possible relays counted implicitly.

Theorem 1. After the preprocessing steps (1)-(3), the algorithm DRC stops after

time 3h, with at most 2h+1 messages passed on by any robot, with all robots know-

ing the minimum di∗o(i∗), the indices i∗ and o(i∗) at which it is attained, and the

orientation of the corresponding tangents. Thus, the total number of messages is

O(h2), with O(h) per robot. Each message size depends only on the encoding size

of coordinate information.

Proof. Full details are omitted due to limited space. To see that the algorithms stops

with the required information as claimed, note that any message D j must have come

through a robot ro( j) opposite to r j after being passed around P once, so any robot


r j receives its own annotated D∗j in h communication rounds after sending out the

unannotated D j. When receiving D∗j for the second time, all D∗

i must have been

encountered, so the current minimum d((ri∗−1,r∗i ),ro(i∗)) is the global minimum.

Even if non-hull relay robots are used, the number of messages per robot remains

O(h); the total number of messages becomes O(ah) for a total of a active robots.

4.2.3 Computing Diameter

The diameter of a polygon is attained between two vertices of the convex hull. We

augment the above algorithm to compute the maximum distance between hull ver-

tices simultaneously by keeping track of the maximum encountered distance in the

vertex descriptions.

5 Results

5.1 Simulation Results

In this section we analyze our algorithms in simulation and compare their perfor-

mance. In the first experiment, we assume there is no error in measurement and we

have enough robots to be placed at the vertices of the object (Fig. 6). As expected,

DRC estimates the exact dimension and orientation for the objects, while the DPCA

estimate is quite good for most of the objects.

We also analyzed DPCA and DRC performances when the number of robots

around the object varies from 4 robots to 45 for the R-shaped object in Fig. 6).

Robots are assumed to be randomly placed at the vertices of the object. The sensors

are assumed with perfect measurement. Fig. 7(top) shows the improvement of DRC

d= 4.37

mw=2.89

theta=1.84

d=3.5

mw=1.29

theta=1.55

d=3.68

mw=3.67

theta=1.88

d=3.25

mw=2.24

theta=1.43

(a)

d=3.91

mw=2.8

theta=1.87

d=3.42

mw =1.15

theta=1.58

d=3.54

mw=3.09

theta=1.55

d=3.23

mw=2.05

theta=1.54

(b)

Fig. 6: Comparison of DPCA(a) and DRC (b) for four different objects. Shown are

object minimum width (solid black), diameter (blue line), orientation (dashed black)

and centroid (blue circle). The convex hull of the objects is shown in red color.

Robots (blue circles) are placed at the vertices of orange objects. The estimates are

also compared quantitatively. The letter d stands for diameter, mw is the minimum

width, theta is the orientation estimate in radians.


and DPCA for estimating the object dimension by increasing the number of robots.

For small numbers of robots, the polygon that is induced by the robots is very differ-

ent from the original object, causing a large error. The mean of the object orientation

error is small even for a small number of robots, because of the symmetric nature

of the orientation value. By picking vertices randomly, the average for orientation

estimation gets close to the actual value. By increasing the number of robots, the

orientation errors of DPCA and DRC tend to zero. Increasing the number of robots

may cause an imbalanced distribution of robots around the object, which affects the

centroid estimation and thus the object dimension estimates. While DRC tends to

picks the closest value to the optimum, which causes it to underestimate dimensions,

DPCA always picks the maximum distance to the centroid, causing it to overesti-

mate.

Number of Robots Around the Object

Err

or(

%)

Minimum Width

0 5 10 15 20 25 30 35 40-100-80

-60-40-20

020

(a)


Err

or(

%)

Diameter

0 5 10 15 20 25 30 35-100

-50

0

50

40

(b)


Err

or(

Rad)

Orientation

DPCADRC

0 5 10 15 20 25 30 35 40-1

-0.5

0

0.5

1

(c)

-100 -50 0 50 100

Minimum Width Error (%)

0

50

100

150

200

250

Fre

qu

en

cy

of

Oc

cu

rre

nc

e DPCA

DRC

(d)

-60 -40 -20 0 20 40 60Diameter Error (%)

0

50

100

150

200

250

300

Fre

qu

en

cy o

f O

ccu

rren

ce

DPCA

DRC

(e)

-2 -1 0 1 2

Orientation Error (Rad)

0

50

100

150

200

250

Fre

qu

en

cy

of

Oc

cu

rre

nc

e DPCA

DRC

(f)

Fig. 7: (Top) The mean and standard deviation of the estimation error for (a) mini-

mum width, (b) diameter, (c) orientation, by DRC (blue) and DPCA (red) for 1000

trials, when the sensors are ideal, but the number of robots varies from m = 4 to

m = 40. The R-shaped object has 30 vertices, with 12 convex vertices. (Bottom)

The almond-shaped object and estimation error distribution of DPCA (red) and

DRC (blue), when sensor errors exist: (d) minimum width (e) diameter (f) object

orientation.

Lastly, we consider the setting in which robots are placed at all vertices of an

almond-shaped object, as shown in Figure 7 (Bottom), and measurement errors ex-

ist. As shown, DRC usually underestimates the minimum width, while DPCA over-

estimates it, for the same reason described above. However, the error of estimating

diameter by DRC has a normal distribution around zero, while DPCA overestimates

the polygon diameter.


5.2 Experimental Results

We used th r-one robot platform [10] to implement DPCA on a real robotic system.

DPCA is used to estimate dimensions and orientation of three different symmetric,

concave and convex objects (Fig. 8). We used 4, 5 and 8 robots to estimate the di-

mension and orientation of rectangle, arrow, and bean-shaped objects, respectively.

In this setup, robots are placed on the vertices of the convex hull of the object.

Robots use our pipelined consensus algorithm to reach the heading consensus and

estimate the object dimension and orientation simultaneously. As shown, DPCA

successfully estimates the object orientation and dimensions with a reasonable er-

ror.

m

(a)

25

15

35

m

(b)

Fig. 8: Experimental result of object characterization by DPCA for three different

objects. 12 trials for each experiment are shown with standard deviation (shadows)

and mean error (solid lines): (a) orientation estimation error (radians). (b) object

diameter (blue) and object minimum width (green) estimation error.

6 Conclusion

We have presented two distributed algorithms for estimating the dimension and ori-

entation of polygonal complex objects. Our algorithms are useful in different ap-

plications in robotics when global sensing is not available. We have tested our al-

gorithms in simulations and experiment. Our algorithms successfully estimate the

dimension and orientation of convex and concave objects. We compared our algo-

rithm in different experiments. While DRC estimates the optimal values when there

are enough robots to sample the object boundary, DPCA is more sensitive to the dis-

tribution of the robots around the object. In the presence of small errors, the more

accurate DRC yields the better results; in the presence of larger errors, DPCA is

to be preferred, as its inherent tendency to overestimate object width may be safer

for avoiding tight corridors in applications of collective object transport. Our algo-

rithms are self-stabilizing and robust to dynamic network topology and population

changes. We will show these features in our future work. One of the future applica-

tions of these algorithms is in collective transport. Manipulator robots estimate the

orientation of the object and adjust the object orientation during motion. Another


potentially interesting direction for future work is to intelligently select subsets of

vertices that lead to accurate estimates of the shapes.

Acknowledgment

We thank several anonymous reviewers for helpful input that improved the presen-

tation of this paper. We also thank Madeleine Nikirk, James Gringe, Sam Caroll,

and Randy Zhang for helping us in data collection.

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